Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

The second boundary value problem in a half-strip for a B-parabolic equation with the Gerasimov–Caputo time derivative

В работе исследуется вторая краевая задача в полуполосе для параболического уравнения с оператором Бесселя, действующим по пространственной переменной, и частной производной Герасимова–Капуто по времени. Доказаны теоремы существования и единственности решения рассматриваемой задачи. Представление решения найдено в терминах интегрального преобразования с функцией Райта в ядре. Единственность решения доказана в классе функций быстрого роста. При частных значениях параметров, содержащихся в рассматриваемом уравнении, последнее совпадает с классическим уравнением диффузии. In the present paper, we investigate the second boundary value problem in a half-strip for a parabolic equation with the Bessel operator acting with respect to the spatial variable and the Gerasimov–Caputo partial time derivative. Theorems of existence and uniqueness of the solution of the problem under consideration are proved.The solution representation is found in terms of an integral transform with the Wright function in the kernel. The uniqueness of the solution is proved in the class of functions of rapid growth. The considered equation for particular values of the parameters coincides with the classical diffusion equation.

Cantor set based neighbor generation method for permutation solution representation

Metaheuristics gained world-wide popularity and researchers have been studying them vigorously in the last two decades. A relatively less explored approach in the improvement of metaheuristics is to design new neighbor generation mechanisms. Neighbor generation mechanisms are very important in the success of any single solution-based heuristic since they directly guide the search. In this study, a neighbor generation mechanism called cantor-set based (CB) method for single solution-based heuristics which use permutation solution representation is described. The inspiration for CB method stems from the recursive algorithm that constructs a cantor set which is a fractal set. Three variations of CB method are discussed (CB-1, CB-2, CB-3) considering the presented design possibilities. The computational experiments are conducted by embedding the mechanisms into the classical local search and simulated annealing algorithms, separately, to test their efficiency and effectiveness by comparing them to classical swap and insertion mechanisms. The traveling salesman problem (TSP) and quadratic assignment problem (QAP) which are very different problems that have incompatible characteristics have been chosen to test the mechanisms and sets of benchmark instances with varying sizes are chosen for the comparisons. The computational tests show that CB-2 gives very favorable results for TSP and CB-1 gives favorable results for QAP which means that CB-2 is suitable for problems that have steep landscapes and CB-1 is suitable for the problems that have flat landscapes. It is observed that CB-3 is a more generalized mechanism because it gives consistently good results for both TSP and QAP instances. The best mechanism for a given instance of the both problem types outperforms the classical neighbor generation of swap and insertion in terms of effectiveness.

Student Understanding of Number Line Graphs

This paper addresses how students understand number line graphs. Utilizing a Think Aloud interview followed by a reflection-eliciting interview, we investigate how two successful College Algebra students understand what it means to graph a statement with one free variable on a number line. These particular students show a mathematically non-normative understanding of this concept; to wit, they do not view the number line graph as representing a solution set. This study illustrates the importance of future research into how students understand the concept of solution representation via number line graphs.